Abstract

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

1. Introduction

Let be a real Hilbert space with inner product and norm , a nonempty closed convex subset of , and the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called strongly positive on if there exists a constant such that
A mapping is called -Lipschitz continuous if there exists a constant such that
In particular, if then is called a nonexpansive mapping; if then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that
The solution set of VIP (3) is denoted by .

The VIP (3) was first discussed by Lions [1] and now is well known; there are a lot of different approaches towards solving VIP (3) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (3) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [2–5]. It is well known that if is strongly monotone and Lipschitz-continuous mapping on , then VIP (3) has a unique solution. Not only are the existence and uniqueness of solutions important topics in the study of VIP (3), but also how to actually find a solution of VIP (3) is important. Up to now, there have been many iterative algorithms in the literature, for finding approximate solutions of VIP (3) and its extended versions; see, for example, [6–11].

In 1976, Korpelevič [12] proposed an iterative algorithm for solving the VIP (3) in Euclidean space :
with a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 11, 13–23] and references therein, to name but a few.

Let be a real-valued function, a nonlinear mapping, and a bifunction. In 2008, Peng and Yao [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that
We denote the set of solutions of GMEP (5) by . The GMEP (5) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied; see, for example, [20, 23–28].

If , then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find such that
It is introduced and studied by S. Takahashi and W. Takahashi [29]. The set of solutions of GEP is denoted by .

If , then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find such that
It is considered and studied in [30–32]. The set of solutions of MEP is denoted by .

If , , then GMEP (5) reduces to the equilibrium problem (EP) which is to find such that
It is considered and studied in [33, 34]. The set of solutions of EP is denoted by . It is worth mentioning that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.

Throughout this paper, we assume as in [18] that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where(H1) for all ;(H2) is monotone; that is, for any ;(H3) is upper-hemicontinuous; that is, for each ,
(H4) is convex and lower semicontinuous for each ;(H5)for each and there exists a bounded subset and such that, for any ,

Given a positive number , let be the solution set of the auxiliary mixed equilibrium problem; that is, for each ,
In particular, whenever , , is rewritten as .

Let be two bifunctions and two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that
where and are two constants. It is introduced and studied in [19]. Whenever , the SGEP reduces to a system of variational inequalities, which is considered and studied in [13]. It is worth mentioning that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [19] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [19]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse strongly monotone for . Let for . Then is a solution of SGEP (12) if and only if is a fixed point of the mapping defined by , where . Here, one denotes the fixed point set of by .

Let be an infinite family of nonexpansive mappings on and a sequence of nonnegative numbers in . For any , define a mapping on as follows:
Such a mapping is called the -mapping generated by and .

In 2011, for the case where , Yao et al. [25] proposed the following hybrid iterative algorithm:
where is a contraction, is differentiable and strongly convex, , and are given, for finding a common element of the set and the fixed point set of an infinite family of nonexpansive mappings on . They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (14) to a point under some appropriate conditions. This point also solves the following optimization problem:
where is the potential function of .

Let be a contraction and a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)–(H4), and that are inverse strongly monotone. Let the mapping be defined as in Proposition CY. Very recently, Ceng et al. [20] introduced the following hybrid extragradient-like iterative algorithm:
for finding a common solution of GMEP (5), SGEP (12), and the fixed point problem of an infinite family of nonexpansive mappings on , where , , , , and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (16) to a point under some suitable conditions. This point also solves the following optimization problem:
where is the potential function of .

On the other hand, let be a nonempty subset of a normed space . A mapping is called uniformly Lipschitzian if there exists a constant such that
Recently, Kim and Xu [35] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as below.

Definition 1. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exist a constant and a sequence in with such that
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with . Subsequently, Sahu et al. [36] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that
Put . Then , , and (13) reduces to the relation
Whenever for all in (21) then is an asymptotically -strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [36] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . More precisely, they first established one weak convergence theorem for the following iterative scheme:
where , , and , and then obtained another strong convergence theorem for the following iterative scheme:
where , , and . Subsequently, the above iterative schemes are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense; see, for example, [10, 22].

In 2009, Yao et al. [30] proposed and analyzed iterative algorithms for finding a common element of the set of fixed points of an asymptotically -strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Very recently, motivated by Yao et al. [30], Cai and Bu [26] introduced and analyzed the following iterative algorithm by the hybrid shrinking projection method:
for finding a common element of the set of solutions of finitely many generalized mixed equilibrium problems, the set of solutions of finitely many variational inequalities for inverse strong monotone mappings , and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense (provided that is nonempty and bounded), where , , , , , . It was proven in [26] that under appropriate conditions converge strongly to .

Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We use the notations and to indicate the weak convergence of to and the strong convergence of to , respectively. Moreover, we use to denote the weak -limit set of ; that is,

Definition 3. A mapping is called(i)monotone if
(ii)-strongly monotone if there exists a constant such that
(iii)-inverse strongly monotone if there exists a constant such that
It is easy to see that the projection is -inverse strongly monotone. The inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 4. A differentiable function is called(i)convex if
where is the Fréchet derivative of at ;(ii)strongly convex if there exists a constant such that
It is easy to see that if is a differentiable strongly convex function with constant then is strongly monotone with constant .

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 5. For given and ,(i), ;(ii), ;(iii), . (This implies that is nonexpansive and monotone.)

By using the technique of [32], we can readily obtain the following elementary result.

Proposition 6 (see [20, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that(i) is strongly convex with constant and the function is weakly upper semicontinuous for each ;(ii)for each and there exists a bounded subset and such that, for any ,
Then the following hold:(a) for each ;(b) is single valued;(c) is nonexpansive if is Lipschitz continuous with constant andwhere for ;(d)for all and ,
(e);(f) is closed and convex.

Remark 7. In Proposition 6, whenever is a bifunction satisfying the conditions (H1)–(H4) and , , we have, for any ,
( is firmly nonexpansive) and
In this case, is rewritten as . If, in addition, , then is rewritten as ; see [19, Lemma 2.1] for more details.

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 8. Let be a real inner product space. Then the following inequality holds:

Lemma 9. Let be a real Hilbert space. Then the following hold:(a) for all ;(b) for all and with ;(c)if is a sequence in such that , it follows that
We have the following crucial lemmas concerning the -mappings defined by (13).

Lemma 10 (see [37, Lemma 3.2]). Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and the limit exists, where is defined by (13).

Lemma 11 (see [37, Lemma 3.3]). Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in for some . Then .

Lemma 12 (see [38, Demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .

Lemma 13. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 5(i)) implies

Lemma 14 (see [36, Lemma 2.5]). Let be a real Hilbert space. Given a nonempty closed convex subset of and points and given also a real number , the set
is convex (and closed).

Recall that a set-valued mapping is called monotone if, for all , and imply
A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Let be a monotone, -Lipschitz-continuous mapping, and let be the normal cone to at ; that is,
Define
Then, is maximal monotone and if and only if ; see [39].

Lemma 15 (see [36, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then
for all and .

Lemma 16 (see [36, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then as .

Lemma 17 (see Demiclosedness principle [36, Proposition 3.1]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 18 (see [36, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.

Lemma 20 (see [43, page 80]). Let , , and be sequences of nonnegative real numbers satisfying the inequality
If and , then exists. If, in addition, has a subsequence which converges to zero, then .

Recall that a Banach space is said to satisfy the Opial condition [38] if, for any given sequence which converges weakly to an element , there holds the inequality
It is well known in [38] that every Hilbert space satisfies the Opial condition.

Lemma 21 (see [22, Proposition 3.1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence in . Suppose that
where and are sequences of nonnegative real numbers such that and . Then converges strongly in .

Lemma 22 (see [44]). Let be a closed convex subset of a real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition
then as .

3. Strong Convergence Theorem

In this section, we will introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions we will prove strong convergence of the proposed algorithm.

Theorem 23. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let , , be three bifunctions from to satisfying (H1)–(H4) and let be a lower semicontinuous and convex functional. Let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively, where and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive mappings on and a sequence in for some . Let be a -strongly positive bounded linear operator with . Let be the -mapping defined by (13). Assume that is nonempty and bounded where is defined as in Proposition CY. Let be a sequence in and , , , and sequences in such that , and . Pick any and set , . Let be a sequence generated by the following algorithm:
where , , , , and , . Assume that the following conditions are satisfied:(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exists a bounded subset and such that, for any ,
(iii) and .Then converges strongly to provided that is firmly nonexpansive.

Proof. As and , we may assume, without loss of generality, that and for all . Since is a -strongly positive bounded linear operator on , we know that
Taking into account that for all , we have
that is, is positive. It follows that
Put
for all , and , where is the identity mapping on . Then we have .We divide the rest of the proof into several steps.Step 1. We show that is well defined. It is obvious that is closed and convex. As the defining inequality in is equivalent to the inequality
by Lemma 14 we know that is convex for every .First of all, let us show that for all . Suppose that for some . Take arbitrarily. Since , is -inverse strongly monotone, and , we have, for any ,
Since , , and is -inverse strongly monotone, where , , by Proposition 5(iii) we deduce that for each
Combining (56) and (57), we have
Since , is -inverse strongly monotone, for , and for , we deduce that, for any ,
(This shows that is nonexpansive.) Also, from (49), (53), (58), and (59) it follows that